Problem:
Algebra-Patterns (InterMath)

Choose four consecutive odd counting numbers. Take the product of the middle two numbers and subtract the product of the first number and the last number. Try and few samples and formulate a rule. Explain why the rule works.

Solution:

Sample #1: 5 7 9 11
(7 x 9) ­ (5 x 11) = 63 ­ 55 = 8

Sample #2: 1 3 5 7
(3 x 5) ­ (1 x 7) = 15-7 = 8

Sample #3: 21 23 25 27
(23 x 25) ­ (21 x 27) = 575 ­ 567 = 8

Okay, so they all equal 8.

1st way:

w x y z
(x * y) ­ (w * z) = 8

w + 2 = x x + 2 = y y + 2 = z
w = x ­ 2

(x * (x+2)) ­ ((x ­ 2)(x + 4))
(x^2 + 2x) ­ (x^2 + 4x ­2x ­8)
(x^2 + 2x) ­ (x^2 + x -8) = 8

Choosing numbers and using guess and check:
If x= 5 35-27 = 8 The rule works!
If x= 21 483-475 = 8 The rule works!

2nd way:

x= Any odd number

Each consecutive odd number you have to add 2 to x, than 4 to x, than 6 to x to get your four consecutive odd counting numbers to work with.
x+ x +2 x+4 x+6

((x + 2) (x +4)) ­ (x (x + 6))
(x^2 + 6x + 8) ­ (x^2 + 6x) = 8

Again, using guess and check to see if the rule works:
x= 5 63-55 = 8 The rule works!
x= 21 575-567= 8 The rule works!